Common Invariant Subspace Research Articles

Computing the structure of finite algebras

In this paper we address some algorithmic problems related to computations in finite-dimensional associative algebras over finite fields. Our starting point is the structure theory of finite-dimensional associative algebras. This theory determines, mostly in a nonconstructive way, the building blocks of these algebras. Our aim is to give polynomial time algorithms to find these building blocks, the radical and the simple direct summands of the radical-free part. The radical algorithm is based on a new, tractable characterisation of the radical. The algorithm for decomposition of semisimple algebras into simple ideals involves (and generalises) factoring polynomials over the ground field. Next, we study the problem of finding zero divisors in finite algebras. We show that thisproblem is in the same complexity class as the problem of factoring polynomials over finte fields. Applications include a polynomial time Las Vegas method to find a common invariant subspace of a set of linear transformations as well as an explicit isomorphism between a given finite simple algebra and a full matrix algebra over a finite field.

ACTION OF BIREGULAR AUTOMORPHISMS OF THE AFFINE PLANE ON PAIRS OF MATRICES

A study is made of the natural action of the group of biregular automorphisms of the affine plane on pairs of square matrices and pairs of symmetric square matrices. It is proved that in the open set of pairs (A, B) such that A is semisimple and A and B have no common invariant subspaces, the commutator [A, B] = AB ? BA is the unique invariant in both cases. Bibliography: 3 titles.

Born's postulate and reconstruction of the?-function in nonrelativistic quantum mechanics

A continuous family of self-adjoint operators is constructed such that their measurement data are insufficient to reproduce uniquely via Born's postulate the underlying quantum state. Moreover, no pair of operators has a common invariant subspace. This rejects a conjecture given by Moroz. On the other hand, strengthening results obtained by Kreinovitch, it is shown that already one special potential and the related localization measurement data at different moments of time can guarantee the uniqueness of reconstruction.

An operator on a Fréchet space with no common invariant subspace with its inverse

A Fréchet space with a two-sided Schauder basis is constructed, such that the corresponding bilateral shift is continuous and invertible, and has no common nontrivial invariant subspace with its inverse. This shows in particular, that the problem of existence of hyperinvariant subspaces for operators on general Fréchet spaces, admits a negative answer. It is also shown that the dual of the Fréchet space constructed can be identified with a commutative locally convex complete topological algebra with unit, which has no closed nontrivial ideals.

A geometric equivalent of the invariant subspace problem

It is shown that every operator has an invariant subspace if and only if every pair of idempotents has a common invariant subspace.